ASYMTOTIC NOTATION ON DATA STRUCTURE AND ALGORITHM

August 3, 2024 Comp 122, Spring 2004
Asymptotic Notation,
Review of Functions &
Summations

Comp 122
ymp - 2
Asymptotic Complexity
 Running time of an algorithm as a function of
input size n for large n.
 Expressed using only the highest-order term in
the expression for the exact running time.
 Instead of exact running time, say (n2
).
 Describes behavior of function in the limit.
 Written using Asymptotic Notation.

Comp 122
ymp - 3
Asymptotic Notation
 , O, , o, 
 Defined for functions over the natural numbers.
 Ex: f(n) = (n2
).
 Describes how f(n) grows in comparison to n2
.
 Define a set of functions; in practice used to compare
two function sizes.
 The notations describe different rate-of-growth
relations between the defining function and the
defined set of functions.

Comp 122
ymp - 4
-notation
(g(n)) = {f(n) :
 positive constants c1, c2, and n0,
such that n  n0,
we have 0  c1g(n)  f(n) 
c2g(n)
}
For function g(n), we define (g(n)),
big-Theta of n, as the set:
g(n) is an asymptotically tight bound for f(n).
Intuitively: Set of all functions that
have the same rate of growth as g(n).

Comp 122
ymp - 5
-notation
(g(n)) = {f(n) :
 positive constants c1, c2, and n0,
we have 0  c1g(n)  f(n) 
c2g(n)
}
For function g(n), we define (g(n)),
big-Theta of n, as the set:
Technically, f(n)  (g(n)).
Older usage, f(n) = (g(n)).
I’ll accept either…
f(n) and g(n) are nonnegative, for large n.

Comp 122
ymp - 6
Example
 10n2
- 3n = (n2
)
 What constants for n0, c1, and c2 will work?
 Make c1 a little smaller than the leading
coefficient, and c2 a little bigger.
 To compare orders of growth, look at the
leading term.
 Exercise: Prove that n2
/2-3n= (n2
)
(g(n)) = {f(n) :  positive constants c1, c2, and n0,
such that n  n0, 0  c1g(n)  f(n)  c2g(n)}

Comp 122
ymp - 7
Example
 Is 3n3
 (n4
) ??
 How about 22n
 (2n
)??
(g(n)) = {f(n) :  positive constants c1, c2, and n0,
such that n  n0, 0  c1g(n)  f(n)  c2g(n)}

Comp 122
ymp - 8
O-notation
O(g(n)) = {f(n) :
 positive constants c and n0,
we have 0  f(n)  cg(n) }
For function g(n), we define O(g(n)),
big-O of n, as the set:
g(n) is an asymptotic upper bound for f(n).
Intuitively: Set of all functions
whose rate of growth is the same as
or lower than that of g(n).
f(n) = (g(n))  f(n) = O(g(n)).
(g(n))  O(g(n)).

Comp 122
ymp - 9
Examples
 Any linear function an + b is in O(n2
). How?
 Show that 3n3
=O(n4
) for appropriate c and n0.
O(g(n)) = {f(n) :  positive constants c and n0,
such that n  n0, we have 0  f(n)  cg(n) }

Comp 122
ymp - 10
 -notation
g(n) is an asymptotic lower bound for f(n).
Intuitively: Set of all functions
whose rate of growth is the same
as or higher than that of g(n).
f(n) = (g(n))  f(n) = (g(n)).
(g(n))  (g(n)).
(g(n)) = {f(n) :
 positive constants c and n0,
we have 0  cg(n)  f(n)}
For function g(n), we define (g(n)),
big-Omega of n, as the set:

Comp 122
ymp - 11
Example
 n = (lg n). Choose c and n0.
(g(n)) = {f(n) :  positive constants c and n0, such
that n  n0, we have 0  cg(n)  f(n)}

Comp 122
ymp - 12
Relations Between , O, 

Comp 122
ymp - 13
Relations Between , , O
 I.e., (g(n)) = O(g(n))  (g(n))
 In practice, asymptotically tight bounds are
obtained from asymptotic upper and lower bounds.
Theorem : For any two functions g(n) and f(n),
f(n) = (g(n)) iff
f(n) = O(g(n)) and f(n) = (g(n)).

Comp 122
ymp - 14
Running Times
 “Running time is O(f(n))”  Worst case is O(f(n))
 O(f(n)) bound on the worst-case running time 
O(f(n)) bound on the running time of every input.
 (f(n)) bound on the worst-case running time 
(f(n)) bound on the running time of every input.
 “Running time is (f(n))”  Best case is (f(n))
 Can still say “Worst-case running time is (f(n))”
 Means worst-case running time is given by some
unspecified function g(n)  (f(n)).

Comp 122
ymp - 15
Example
 Insertion sort takes (n2
) in the worst case, so
sorting (as a problem) is O(n2
). Why?
 Any sort algorithm must look at each item, so
sorting is (n).
 In fact, using (e.g.) merge sort, sorting is (n lg n)
in the worst case.
 Later, we will prove that we cannot hope that any
comparison sort to do better in the worst case.

Comp 122
ymp - 16
Asymptotic Notation in Equations
 Can use asymptotic notation in equations to
replace expressions containing lower-order terms.
 For example,
4n3
+ 3n2
+ 2n + 1 = 4n3
+ 3n2
+ (n)
= 4n3
+ (n2
) = (n3
). How to interpret?
 In equations, (f(n)) always stands for an
anonymous function g(n)  (f(n))
 In the example above, (n2
) stands for
3n2
+ 2n + 1.

Comp 122
ymp - 17
o-notation
f(n) becomes insignificant relative to g(n) as n
approaches infinity:
lim [f(n) / g(n)] = 0
n
g(n) is an upper bound for f(n) that is not
asymptotically tight.
Observe the difference in this definition from previous
ones. Why?
o(g(n)) = {f(n):  c > 0,  n0 > 0 such that
 n  n0, we have 0  f(n) < cg(n)}.
For a given function g(n), the set little-o:

Comp 122
ymp - 18
(g(n)) = {f(n):  c > 0,  n0 > 0 such that
 n  n0, we have 0  cg(n) < f(n)}.
 -notation
f(n) becomes arbitrarily large relative to g(n) as n
approaches infinity:
lim [f(n) / g(n)] = .
n
g(n) is a lower bound for f(n) that is not
asymptotically tight.
For a given function g(n), the set little-omega:

Comp 122
ymp - 19
Comparison of Functions
f  g  a  b
f (n) = O(g(n))  a  b
f (n) = (g(n))  a  b
f (n) = (g(n))  a = b
f (n) = o(g(n))  a < b
f (n) = (g(n))  a > b

Comp 122
ymp - 20
Limits
 lim [f(n) / g(n)] = 0  f(n)  (g(n))
n
 lim [f(n) / g(n)] <   f(n)  (g(n))
n
 0 < lim [f(n) / g(n)] <   f(n)  (g(n))
n
 0 < lim [f(n) / g(n)]  f(n) (g(n))
n
 lim [f(n) / g(n)] =   f(n)  (g(n))
n
 lim [f(n) / g(n)] undefined can’t say
n

Comp 122
ymp - 21
Properties
 Transitivity
f(n) = (g(n)) & g(n) = (h(n))  f(n) = (h(n))
f(n) = O(g(n)) & g(n) = O(h(n))  f(n) = O(h(n))
f(n) = (g(n)) & g(n) = (h(n))  f(n) = (h(n))
f(n) = o (g(n)) & g(n) = o (h(n))  f(n) = o (h(n))
f(n) = (g(n)) & g(n) = (h(n))  f(n) = (h(n))
 Reflexivity
f(n) = (f(n))
f(n) = O(f(n))
f(n) = (f(n))

Comp 122
ymp - 22
Properties
 Symmetry
f(n) = (g(n)) iff g(n) = (f(n))
 Complementarity
f(n) = O(g(n)) iff g(n) = (f(n))
f(n) = o(g(n)) iff g(n) = ((f(n))

Common Functions

Comp 122
ymp - 24
Monotonicity
 f(n) is
 monotonically increasing if m  n  f(m)  f(n).
 monotonically decreasing if m  n  f(m)  f(n).
 strictly increasing if m < n  f(m) < f(n).
 strictly decreasing if m > n  f(m) > f(n).

Comp 122
ymp - 25
Exponentials
 Useful Identities:
 Exponentials and polynomials
n
m
n
m
mn
n
m
a
a
a
a
a
a
a





)
(
1
1
)
(
0
lim
n
b
n
b
n
a
o
n
a
n






Comp 122
ymp - 26
Logarithms
x = logba is the
exponent for a = bx
.
Natural log: ln a = logea
Binary log: lg a = log2a
lg2
a = (lg a)2
lg lg a = lg (lg a)
a
c
a
b
b
b
c
c
b
b
n
b
c
c
c
a
b
b
b
c
a
b
a
a
a
b
a
a
a
n
a
b
a
ab
b
a
log
log
log
log
1
log
log
)
/
1
(
log
log
log
log
log
log
log
log
)
(
log










Comp 122
ymp - 27
Logarithms and exponentials – Bases
 If the base of a logarithm is changed from one
constant to another, the value is altered by a
constant factor.
 Ex: log10 n * log210 = log2 n.
 Base of logarithm is not an issue in asymptotic
notation.
 Exponentials with different bases differ by a
exponential factor (not a constant factor).
 Ex: 2n
= (2/3)n
*3n
.

Comp 122
ymp - 28
Polylogarithms
 For a  0, b > 0, lim n ( lga
n / nb
) = 0,
so lga
n = o(nb
), and nb
= (lga
n )
 Prove using L’Hopital’s rule repeatedly
 lg(n!) = (n lg n)
 Prove using Stirling’s approximation (in the text) for lg(n!).

Comp 122
ymp - 29
Exercise
Express functions in A in asymptotic notation using functions in B.
A B
5n2
+ 100n 3n2
+ 2
A  (n2
), n2
 (B)  A  (B)
log3(n2
) log2(n3
)
logba = logca / logcb; A = 2lgn / lg3, B = 3lgn, A/B =2/(3lg3)
nlg4
3lg n
alog b
=blog a
; B =3lg n
=nlg 3
; A/B =nlg(4/3)
 as n
lg2
n n1/2
lim ( lga
n / nb
) = 0 (here a = 2 and b = 1/2)  A  (B)
n
A  (B)
A  (B)
A  (B)
A  (B)

Summations – Review

Comp 122
ymp - 31
Review on Summations
 Why do we need summation formulas?
For computing the running times of iterative
constructs (loops). (CLRS – Appendix A)
Example: Maximum Subvector
Given an array A[1…n] of numeric values (can be
positive, zero, and negative) determine the
subvector A[i…j] (1 i  j  n) whose sum of
elements is maximum over all subvectors.
1 -2 2 2

Comp 122
ymp - 32
MaxSubvector(A, n)
maxsum  0;
for i  1 to n
do for j = i to n
sum  0
for k  i to j
do sum += A[k]
maxsum  max(sum, maxsum)
return maxsum
n n j
T(n) =    1
i=1 j=i k=i
NOTE: This is not a simplified solution. What is the final answer?

Comp 122
ymp - 33
 Constant Series: For integers a and b, a  b,
 Linear Series (Arithmetic Series): For n  0,
 Quadratic Series: For n  0,





b
a
i
a
b 1
1
2
)
1
(
2
1
1








n
n
n
i
n
i










n
i
n
n
n
n
i
1
2
2
2
2
6
)
1
2
)(
1
(
2
1 

Comp 122
ymp - 34
 Cubic Series: For n  0,
 Geometric Series: For real x  1,
For |x| < 1,








n
i
n
n
n
i
1
2
2
3
3
3
3
4
)
1
(
2
1 











n
k
n
n
k
x
x
x
x
x
x
0
1
2
1
1
1 


 

0 1
1
k
k
x
x

Comp 122
ymp - 35
 Linear-Geometric Series: For n  0, real c  1,
 Harmonic Series: nth harmonic number, nI+
,














n
i
n
n
n
i
c
c
nc
c
n
nc
c
c
ic
1
2
2
1
2
)
1
(
)
1
(
2 
n
Hn
1
3
1
2
1
1 



 





n
k
O
n
k
1
)
1
(
)
ln(
1

Comp 122
ymp - 36
 Telescoping Series:
 Differentiating Series: For |x| < 1,


 


n
k
n
k
k a
a
a
a
1
0
1
 


 

0
2
1
k
k
x
x
kx

Comp 122
ymp - 37
 Approximation by integrals:
 For monotonically increasing f(n)
 For monotonically decreasing f(n)
 How?
  
 



n
m
n
m
k
n
m
dx
x
f
k
f
dx
x
f
1
1
)
(
)
(
)
(
  

 


1
1
)
(
)
(
)
(
n
m
n
m
k
n
m
dx
x
f
k
f
dx
x
f

Comp 122
ymp - 38
 nth harmonic number
 





n
k
n
n
x
dx
k
1
1
1
)
1
ln(
1
 



n
k
n
n
x
dx
k
2 1
ln
1





n
k
n
k
1
1
ln
1

Comp 122
ymp - 39
Reading Assignment
 Chapter 4 of CLRS.

ASYMTOTIC NOTATION ON DATA STRUCTURE AND ALGORITHM

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